Algebra problem (from lang's basic mathematics)

Hello everyone,

I've recently decided to try and brush up on my math by self-studying Lang's basic mathematics. There's an exercise there which reads: "Justify each step, using commutativity and associativity in proving the following identities". I've solved 8/10 problems but the last two I can't seem to figure out:

9. (X - Y) - (Z -W) = (X + W) - Y -Z

and:

10. (X - Y) - (Z -W) = (X - Z) + (W - Y)

Before the problems you're given the principle of commutativity and associativity, as well as the identity: "-(A + B) = - A - B", which I assume are all relevant in solving these problems.

If I replace all the variables with numbers I can see clearly that they are identical, for some reason however I cannot translate that into a series of abstract steps that proves that these are identical. Any help would be greatly appreciated!

Re: Algebra problem (from lang's basic mathematics)

Back in 1930 the rule was "When removing brackets preceded by a minus sign change the signsof the terms within the brackets" There is a 1 understood after the minus so -1 must be multiplied by each term

Re: Algebra problem (from lang's basic mathematics)

Thank you very much both. It wasn't explained that you could do that; once you point it out it seems obvious though. I'm glad it wasn't such a simple solution as I thought it would be, because I felt quite stupid to get stuck in the first chapter like that